Throughout this section we take \(p_{\lambda }\)  to be the SCAD penalty defined in (REF ), although the results we present generalize to other penalty functions including the Lasso and MCP. We note that while the general framework (REF ) allows a penalty on every element and pairwise difference of the regression parameter vectors \(\beta _1\) , \(\beta _2\) , and \(\beta _3\) , the estimator based on (REF ) specifically penalizes \(\beta _1\)  and its pairwise differences with \(\beta _2\)  and \(\beta _3\) . This facilitates the theoretical analysis while retaining the property that the elementwise differences between \(\beta _g\) 's are sparse. Moreover, to accommodate the role of nonconvexity the constraint \(\Vert \psi \Vert _{1} \le {} R_1\)  is imposed on the parameter space over which solutions are sought [1]}. See Assumption REF  below and its remark for detailed discussion.
